Question: Emeka and Maricel were asked to find an explicit formula for the sequence $79,71,63,55,...$, where the first term should be $h(1)$. Emeka said the formula is $h(n)=79-8(n-1)$. Maricel said the formula is $h(n)=87-8n$. Which one of them is right? Choose 1 answer: Choose 1 answer: (Choice A) A Only Emeka (Choice B) B Only Maricel (Choice C) C Both Emeka and Maricel (Choice D) D Neither Emeka nor Maricel
Solution: The general explicit formula for arithmetic sequences is ${a_1}+{d}(n-1)$, where ${a_1}$ is the first term and $ d$ is the common difference. The first term is ${79}$ and the common difference is ${-8}$. ${-8\,\curvearrowright}$ ${-8\,\curvearrowright}$ ${-8\,\curvearrowright}$ ${79},$ $71,$ $63,$ $55,...$ We get the following formula. $h(n)={79}{-8}(n-1)$ So Emeka is definitely right. What about Maricel? We can see that in Maricel's formula, the constant difference is multiplied by $n$ and not by $(n-1)$. Let's expand the parentheses in Emeka's formula to arrive at a similar expression form: $\begin{aligned} h(n)&=79-8(n-1)\\\\ &=79-8n+8\\\\ &=87-8n\end{aligned}$ We obtained Maricel's formula, which means it's also a correct explicit formula for $h(n)$. Both Emeka and Maricel got a correct explicit formula.